Error estimates for a mixed finite element method for the Maxwell's transmission eigenvalue problem (Q6593975)

The presented approach for solving the Maxwell's transmission eigenvalue problem is based on an approach proposed in [\textit{J. Sun} and \textit{L. Xu}, Inverse Probl. 29, No. 10, Article ID 104013, 18 p. (2013; Zbl 1286.65149)]. At first the Maxwell's transmission eigenvalue problem is reformulated as a nonlinear non-selfadjoint quad-curl problem. The real transmission eigenvalues are the roots of a nonlinear algebraic equation related to a series of self-adjoint positive definite quad-curl eigenvalue problems. A mixed formulation for the quad-curl eigenvalue problems is given. For the discretization of these problems the curl conforming edge elements of Nédélec are used. For the computation of the roots of the nonlinear equation an algorithm is proposed which is based on a secant method. A convergence analysis is given, where the convergence of the mixed finite element method is proved. An error estimate of the iterative algorithm for the transmission eigenvalue is established.